one day you go hiking at a nearby nature preserve. At first, you follow the stra

Question

one day you go hiking at a nearby nature preserve. At first, you follow the straight, clearly marked trails. From your starting point, you travel 2.00 miles down the 1st trail. Then you turn to your left by 30.0 degrees to follow a 2nd trial for 1.00 miles. Next, you turn to your right by 160 degrees and follow a 3rd trail for 2.20 miles. At this point you are getting very tired and would like to get back as quickly as possible, but all the available trails seem to lead you deeper into the woods. You would like to take a shortcut directly through the woods (ignoring the trails). How far to your right should you turn, and how far do you have to walk, to go directly back to your starting point??

Explanation / Answer

Let’s say that the initial direction is east. 1st you walked 2 miles east. To turn 30° left of east, you rotate your body 30° counter clockwise. Now you are walking 30° north of east. East component = 1.2 * cos 30° North component = 1.2 * sin 30° 2nd you walked (1.2 * cos 30°) miles east and (1.2 * sin 30°) miles north. To turn 160° right, you rotate your body 160° clockwise. Let’s add these 2 angles. 30° counter clockwise + 160° clockwise = 130° clockwise. 130° clockwise from east is 50° south of west. West component = 1.80 * cos 50° South component = 1.80 * sin 50° Let’s determine your final position. 2 miles east + 1.2 * cos 30° miles east + 1.80 * cos 50° miles west = 1.8822 miles east 1.2 * sin 30° miles north + 1.80 * sin 50° miles south = 0.779 mile south Your final position is 1.8822 miles east and 0.779 mile south of your initial position. Total distance walked = (1.8822^2 + 0.779^2)^0.5 = 2.037 miles The tangent of angle south of east = 0.779 ÷ 1.8822 Angle south of east = 22.5° You walked 2.037 miles at an angle of 22.5° south of east. To return to your initial position, you must walk 2.037 miles in the opposite direction. Opposite direction = 22.5° north of west. As you walk 2.037 miles at an angle of 22.5° north of west, you will walk 1.8822 miles west and 0.779 mile north. This should put you back to your original position.

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